Animation moves armature
With an object selected, go into the Object sub-panel of the Properties Editor. Under the Relations section is a Parent field where you can click an x to get rid of a parent or use the eyedropper to choose a parent.

Select the armature's root bone then go to the Constraints sub-panel to see if there is a constraint that could be causing the problem.

The Constraint sub-panel is (in blender 2.79) in the Properties Editor, 6th icon along. See picture: